Frequency combs based on ultrafast lasers and other frequency comb sources are immensely powerful tools for a wide range of applications. Exemplary frequency comb sources include mode-locked lasers, which generate optical frequency combs that, when broadened, can span octave bandwidths. An optical frequency comb appears in the time domain as an optical pulse train, which can be characterized as a high-frequency carrier modulated by an envelope that pulses at a predetermined repetition rate frep. The repetition rate equals the speed of light in the laser cavity divided by the round-trip length of the laser cavity and may be on the order of Megahertz or Gigahertz rates. Each pulse in the pulse train can be as brief as one to several optical cycles, e.g., pulse widths of picoseconds, femtoseconds, or even attoseconds.
FIG. 1 shows a time-domain representation (top) and a frequency-domain representation (bottom) of an exemplary optical frequency comb. In the time domain, the comb appears as a narrow Gaussian envelope (dotted line 10) that modulates the amplitude of a carrier oscillating at the laser frequency (solid line 12). In the frequency domain, the Fourier transform of the envelope is a broad Gaussian 20 that modulates the amplitude of the frequency comb, which has orders 22 spaced at the repetition frequency, frep.
The frequency comb 22 in FIG. 1 can also be characterized by its carrier-envelope phase (or absolute phase), which is the phase of the envelope with respect to the phase of the carrier. In mode-locked lasers, the carrier-envelope phase may vary as function of time, e.g., it may advance or recede with successive pulses in the pulse train. If the carrier-envelope phase changes by Δφ with each successive pulse, as shown in FIG. 1, then the carrier-envelope offset (CEO) frequency can be expressed as:
      f    CEO    =            Δφmod2π              2        ⁢        π              ⁢                  f        rep            .      The CEO frequency appears in the frequency domain as a displacement of the frequency comb from dc. Without stabilization, the carrier-envelope phase may drift as the laser cavity changes temperature, vibrates, changes refractive index, or experiences other environmental or optical perturbations.